Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,15,Mod(17,18)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.17");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(22.3792142673\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 44875x^{2} + 44876x + 503643366 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 2x^{3} - 44875x^{2} + 44876x + 503643366 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 44867\nu + 22434 ) / 89769 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -864\nu^{3} + 1296\nu^{2} + 58162752\nu - 29081592 ) / 29923 \)
|
| \(\beta_{3}\) | \(=\) |
\( 324\nu^{2} - 324\nu - 7269912 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 1296\beta _1 + 648 ) / 1296 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 1296\beta _1 + 29080296 ) / 1296 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 22435\beta_{2} + 87246072\beta _1 + 43620120 ) / 1296 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
− | 90.5097i | 0 | −8192.00 | − | 132922.i | 0 | 1.63562e6 | 741455.i | 0 | −1.20307e7 | ||||||||||||||||||||||||||||
| 17.2 | − | 90.5097i | 0 | −8192.00 | 141636.i | 0 | −305793. | 741455.i | 0 | 1.28194e7 | ||||||||||||||||||||||||||||||
| 17.3 | 90.5097i | 0 | −8192.00 | − | 141636.i | 0 | −305793. | − | 741455.i | 0 | 1.28194e7 | |||||||||||||||||||||||||||||
| 17.4 | 90.5097i | 0 | −8192.00 | 132922.i | 0 | 1.63562e6 | − | 741455.i | 0 | −1.20307e7 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 18.15.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 18.15.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 144.15.e.b | 4 | ||
| 12.b | even | 2 | 1 | 144.15.e.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.15.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 18.15.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 144.15.e.b | 4 | 4.b | odd | 2 | 1 | ||
| 144.15.e.b | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 37728973188T_{5}^{2} + 354437717876817322500 \)
acting on \(S_{15}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8192)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} - 1329832 T - 500161786544)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 64\!\cdots\!84 \)
|
| $13$ |
\( (T^{2} + \cdots - 24\!\cdots\!76)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 21\!\cdots\!84 \)
|
| $19$ |
\( (T^{2} + \cdots - 35\!\cdots\!84)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 18\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 16\!\cdots\!64 \)
|
| $31$ |
\( (T^{2} + \cdots + 65\!\cdots\!16)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots + 97\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 13\!\cdots\!64 \)
|
| $43$ |
\( (T^{2} + \cdots - 23\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 89\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 50\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 41\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 95\!\cdots\!04)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 27\!\cdots\!04 \)
|
| $73$ |
\( (T^{2} + \cdots + 85\!\cdots\!36)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 19\!\cdots\!44)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 57\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 16\!\cdots\!44 \)
|
| $97$ |
\( (T^{2} + \cdots - 38\!\cdots\!24)^{2} \)
|
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